Online Universal Search in Proof Space

BIOPS starts with a probability distribution
(the initial bias) on the proof techniques that
one can write in ,
e.g.,
for programs composed from
possible instructions [26].
BIOPS is near-bias-optimal
[47]
in the sense that it will not spend
much more time on any proof technique than it deserves,
according to its probabilistic bias,
namely, not much more than its probability times the total search time:

Definition 5.1 (Bias-Optimal
Searchers [47])
Let be a problem class,
be a search space of solution candidates
(where any problem should have a solution in ),
be a task-dependent bias in the form of conditional probability
distributions on the candidates . Suppose that we also have
a predefined procedure that creates and tests any given
on any within time (typically unknown in advance).
Then a searcher is -bias-optimal () if
for any maximal total search time
it is guaranteed to solve any problem
if it has a solution
satisfying
.
It is bias-optimal if .

Undo effects of on
(does not cost significantly more time
than executing ).

A proof technique can interrupt Method 5.1
only by invoking instruction check() (Item 5),
which may transfer
control to switchprog (which possibly
even will delete or rewrite Method 5.1).
Since the initial
runs on the formalized hardware, and since proof techniques
tested by can read and other parts of , they
can produce proofs concerning the (expected)
performance of and BIOPS itself.
Method 5.1 at least has the
optimal order of computational complexity in the following
sense.

Theorem 5.1
If independently of variable time(s) some unknown
fast proof technique
would require at most steps to produce
a proof of difficulty measure (an integer depending on
the nature of the task to be solved), then
Method 5.1
will need at most steps.

Proof.
It is easy to see that
Method 5.1 will need at most
steps--the constant factor does not depend on .
Q.E.D.

Note again, however, that
the proofs themselves may concern quite
different, arbitrary formalizable notions of optimality
(stronger than those expressible in the -notation)
embodied by the given, problem-specific, formalized
utility function .
This may provoke useful, constant-affecting rewrites of
the initial proof searcher despite its limited (yet popular
and widely used)
notion of -optimality.